Use the general slicing method to find the volume of the following solids. the solid whose base is the region bounded by the curves y=x² and y=2−x², and whose cross sections through the solid perpendicular to the x-axis are squares

The volume is

Step-by-step explanation:

The General Slicing Method is given by

Suppose a solid object extends from x = a to x = b and the cross section of the solid perpendicular to the x-axis has an area given by a function A that is integrable on [a, b]. The volume of the solid is

Because a typical cross section perpendicular to the x-axis is a square disk (according with the graph below), the area of a cross section is

The key observation is that the width is the distance between the upper bounding curve and the lower bounding curve

The width of each square is given by

This means that the area of the square cross section at the point x is

The intersection points of the two bounding curves satisfy , which has solutions x = ±1.

Therefore, the cross sections lie between x = -1 and x = 1. Integrating the cross-sectional areas, the volume of the solid is

you need a common difference between consecutive terms to be constant, like 1,2,3,4 or 2,4,6,8, or 1,3,6,9

i found two solutions,   n = 2 • ± √3 = ± 3.4641

step-by-step explanation: